The Generalized Ricci Flow for 3D Manifolds with One Killing Vector

We consider 3D flow equations inspired by the renormalization group (RG) equations of string theory with a three dimensional target space. By modifying the flow equations to include a U(1) gauge field, and adding carefully chosen De Turck terms, we are able to extend recent 2D results of Bakas to the case of a 3D Riemannian metric with one Killing vector. In particular, we show that the RG flow with De Turck terms can be reduced to two equations: the continual Toda flow solved by Bakas, plus its linearizaton. We find exact solutions which flow to homogeneous but not always isotropic geometries

Radiation exposure awareness from patients undergoing nuclear medicine diagnostic 99mTc-MDP bone scans and 2-deoxy-2-(18F) fluoro-D-glucose PET/computed tomography scans

INTRODUCTION: Medical imaging is on average the largest source of artificial radiation exposure worldwide. This study seeks to understand patient's awareness of radiation exposure derived from nuclear medicine diagnostic scans and assess if current information provided by leaflets is adequate. METHODS: Single-centre cross-sectional questionnaire study applied to bone scan and FDG PET/computed tomography patients, at a nuclear medicine and PET/computed tomography department over a 15-week period in 2018. Questionnaires on dose comparators were designed in collaboration with patients, public, and experts in radiation exposure. Qualitative data were analysed using thematic analysis and quantitative data using SPSS (V. 24). RESULTS: A total of 102 questionnaires were completed (bone scan = 50; FDG PET/computed tomography = 52). Across both groups, 33/102 (32.4%) patients reported having a reasonable understanding of nuclear medicine and 21/102 (20.6%) reported a reasonable knowledge of ionising radiations. When asked to compare the exposure dose of respective scans with common comparators 8/50 (16%) of bone scan patients and 11/52 (21.2%) FDG PET/computed tomography answered correctly. On leaflet information, 15/85 (17.6%) patients reported the leaflets do not provide enough information on radiation exposure and of these 10/15 (66.7%) commented the leaflets should incorporate more information on radiation exposure dose. CONCLUSION: More observational and qualitative studies in collaboration with patients are warranted to evaluate patients' understanding and preferences in communication of radiation exposure from nuclear medicine imaging. This will ensure communication tools and guidelines developed to comply with ionising radiation (medical exposure) regulation 2017 are according to patients needs and preferences

Learning SO(3) Equivariant Representations with Spherical CNNs

We address the problem of 3D rotation equivariance in convolutional neural networks. 3D rotations have been a challenging nuisance in 3D classification tasks requiring higher capacity and extended data augmentation in order to tackle it. We model 3D data with multi-valued spherical functions and we propose a novel spherical convolutional network that implements exact convolutions on the sphere by realizing them in the spherical harmonic domain. Resulting filters have local symmetry and are localized by enforcing smooth spectra. We apply a novel pooling on the spectral domain and our operations are independent of the underlying spherical resolution throughout the network. We show that networks with much lower capacity and without requiring data augmentation can exhibit performance comparable to the state of the art in standard retrieval and classification benchmarks.Comment: Camera-ready. Accepted to ECCV'18 as oral presentatio

Holography for the Lorentz Group Racah Coefficients

A known realization of the Lorentz group Racah coefficients is given by an integral of a product of 6 ``propagators'' over 4 copies of the hyperbolic space. These are ``bulk-to-bulk'' propagators in that they are functions of two points in the hyperbolic space. It is known that the bulk-to-bulk propagator can be constructed out of two bulk-to-boundary ones. We point out that there is another way to obtain the same object. Namely, one can use two bulk-to-boundary and one boundary-to-boundary propagator. Starting from this construction and carrying out the bulk integrals we obtain a realization of the Racah coefficients that is ``holographic'' in the sense that it only involves boundary objects. This holographic realization admits a geometric interpretation in terms of an ``extended'' tetrahedron.Comment: 12 pages, 2 figures; v2: minor changes; v3: "extended" tetrahedron interpretation adde

Quantum creation of an Inhomogeneous universe

In this paper we study a class of inhomogeneous cosmological models which is a modified version of what is usually called the Lema\^itre-Tolman model. We assume that we have a space with 2-dimensional locally homogeneous spacelike surfaces. In addition we assume they are compact. Classically we investigate both homogeneous and inhomogeneous spacetimes which this model describe. For instance one is a quotient of the AdS$_4$ space which resembles the BTZ black hole in AdS$_3$. Due to the complexity of the model we indicate a simpler model which can be quantized easily. This model still has the feature that it is in general inhomogeneous. How this model could describe a spontaneous creation of a universe through a tunneling event is emphasized.Comment: 21 pages, 5 ps figures, REVTeX, new subsection include

The late-time behaviour of vortic Bianchi type VIII Universes

We use the dynamical systems approach to investigate the Bianchi type VIII models with a tilted $\gamma$-law perfect fluid. We introduce expansion-normalised variables and investigate the late-time asymptotic behaviour of the models and determine the late-time asymptotic states. For the Bianchi type VIII models the state space is unbounded and consequently, for all non-inflationary perfect fluids, one of the curvature variables grows without bound. Moreover, we show that for fluids stiffer than dust ($1<\gamma<2$), the fluid will in general tend towards a state of extreme tilt. For dust ($\gamma=1$), or for fluids less stiff than dust ($0<\gamma< 1$), we show that the fluid will in the future be asymptotically non-tilted. Furthermore, we show that for all $\gamma\geq 1$ the universe evolves towards a vacuum state but does so rather slowly, $\rho/H^2\propto 1/\ln t$.Comment: 19 pages, 3 ps figures, v2:typos fixed, refs and more discussion adde

Light scattering and phase behavior of Lysozyme-PEG mixtures

Measurements of liquid-liquid phase transition temperatures (cloud points) of mixtures of a protein (lysozyme) and a polymer, poly(ethylene glycol) (PEG) show that the addition of low molecular weight PEG stabilizes the mixture whereas high molecular weight PEG was destabilizing. We demonstrate that this behavior is inconsistent with an entropic depletion interaction between lysozyme and PEG and suggest that an energetic attraction between lysozyme and PEG is responsible. In order to independently characterize the lysozyme/PEG interactions, light scattering experiments on the same mixtures were performed to measure second and third virial coefficients. These measurements indicate that PEG induces repulsion between lysozyme molecules, contrary to the depletion prediction. Furthermore, it is shown that third virial terms must be included in the mixture's free energy in order to qualitatively capture our cloud point and light scattering data. The light scattering results were consistent with the cloud point measurements and indicate that attractions do exist between lysozyme and PEG.Comment: 5 pages, 2 figures, 1 tabl

Cohomology of groups of diffeomorphims related to the modules of differential operators on a smooth manifold

Let $M$ be a manifold and $T^*M$ be the cotangent bundle. We introduce a 1-cocycle on the group of diffeomorphisms of $M$ with values in the space of linear differential operators acting on $C^{\infty} (T^*M).$ When $M$ is the $n$-dimensional sphere, $S^n$, we use this 1-cocycle to compute the first-cohomology group of the group of diffeomorphisms of $S^n$, with coefficients in the space of linear differential operators acting on contravariant tensor fields.Comment: arxiv version is already officia

Diffusion on a heptagonal lattice

We study the diffusion phenomena on the negatively curved surface made up of congruent heptagons. Unlike the usual two-dimensional plane, this structure makes the boundary increase exponentially with the distance from the center, and hence the displacement of a classical random walker increases linearly in time. The diffusion of a quantum particle put on the heptagonal lattice is also studied in the framework of the tight-binding model Hamiltonian, and we again find the linear diffusion like the classical random walk. A comparison with diffusion on complex networks is also made.Comment: 5 pages, 6 figure

On the Topology of Black Hole Event Horizons in Higher Dimensions

In four dimensions the topology of the event horizon of an asymptotically flat stationary black hole is uniquely determined to be the two-sphere $S^2$. We consider the topology of event horizons in higher dimensions. First, we reconsider Hawking's theorem and show that the integrated Ricci scalar curvature with respect to the induced metric on the event horizon is positive also in higher dimensions. Using this and Thurston's geometric types classification of three-manifolds, we find that the only possible geometric types of event horizons in five dimensions are $S^3$ and $S^2 \times S^1$. In six dimensions we use the requirement that the horizon is cobordant to a four-sphere (topological censorship), Friedman's classification of topological four-manifolds and Donaldson's results on smooth four-manifolds, and show that simply connected event horizons are homeomorphic to $S^4$ or $S^2\times S^2$. We find allowed non-simply connected event horizons $S^3\times S^1$ and $S^2\times \Sigma_g$, and event horizons with finite non-abelian first homotopy group, whose universal cover is $S^4$. Finally, following Smale's results we discuss the classification in dimensions higher than six.Comment: 12 pages, minor edits 27/09/0